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Saturation Problems for Families of Automata

Authors León Bohn , Yong Li , Christof Löding , Sven Schewe

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LIPIcs.ICALP.2025.146.pdf
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  • Theory of computation → Automata over infinite objects
Keywords
  • Families of Automata
  • automata learning
  • FDFAs

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Abstract

Families of deterministic finite automata (FDFA) represent regular ω-languages through their ultimately periodic words (UP-words). An FDFA accepts pairs of words, where the first component corresponds to a prefix of the UP-word, and the second component represents a period of that UP-word. An FDFA is termed saturated if, for each UP-word, either all or none of the pairs representing that UP-word are accepted. We demonstrate that determining whether a given FDFA is saturated can be accomplished in polynomial time, thus improving the known PSPACE upper bound by an exponential. We illustrate the application of this result by presenting the first polynomial learning algorithms for representations of the class of all regular ω-languages. Furthermore, we establish that deciding a weaker property, referred to as almost saturation, is PSPACE-complete. Since FDFAs do not necessarily define regular ω-languages when they are not saturated, we also address the regularity problem and show that it is PSPACE-complete. Finally, we explore a variant of FDFAs called families of deterministic weak automata (FDWA), where the semantics for the periodic part of the UP-word considers ω-words instead of finite words. We demonstrate that saturation for FDWAs is also decidable in polynomial time, that FDWAs always define regular ω-languages, and we compare the succinctness of these different models.

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León Bohn, Yong Li, Christof Löding, and Sven Schewe. Saturation Problems for Families of Automata. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 146:1-146:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.146

Author Details

León Bohn
  • RWTH Aachen University, Germany
Yong Li
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
Christof Löding
  • RWTH Aachen University, Germany
Sven Schewe
  • University of Liverpool, UK

Funding

  • Bohn, León: Supported by DFG grant LO 1174/7-1.
  • Li, Yong: Supported by National Natural Science Foundation of China (Grant No. 62102407).
  • Schewe, Sven: Supported by the EPSRC through projects EP/X03688X/1 and EP/X042596/1.

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